Given any admissible -dimensional family of immersions of a given closed oriented surface into an arbitrary closed Riemannian manifold, we prove that the corresponding min-max width for the area is achieved by a smooth (possibly branched) immersed minimal surface with multiplicity and Morse index bounded by .
Alessandro Pigati. Tristan Rivière. "A proof of the multiplicity conjecture for min-max minimal surfaces in arbitrary codimension." Duke Math. J. 169 (11) 2005 - 2044, 15 August 2020. https://doi.org/10.1215/00127094-2020-0002